Dear Tim,
> You have not really addressed my crucial question: How do you rule out the possibility that your proposed experiment, instead of giving us new mathematical knowledge, simply serves to disprove the physical theory?
We rule this out because in the past, we have never encountered such a thing. Can you give an example of an experiment which has predicted new mathematical knowledge (along the line of "ZFC is consistent" or "CH is false") and was discarded because it is at odds with a physical theory which later turned out to be incorrect?
Now it is indeed the case that we cannot rule out what your are suggesting. It just seems very unlikely (especially to a non-logician)
and we have never encountered such a phenomenon. Therefore, a physicist would certainly not consider it, as it would not add to his enterprise, the modelling and predicting of physical reality.
> The answer, of course, is that you can't. All you can do is stubbornly insist that you believe the physical theory.
A theory which keeps producing verifiable predictions, which usually turn out to be correct, or correct after some adjustment to the theory.
> This is no different in principle from stubbornly insisting *right now* that ZFC is consistent. In both cases you're just looking at a finite amount of evidence and dogmatically extrapolating outside well-confirmed parameter ranges.
There *is* a difference: the physical world cannot be whatever; there are no "infinite possibilities". There is a (very large but finite) upper bound on the number of (reasonable) scenarios of e.g. how the earth will develop. For instance, when the sun goes supernova, the earth will be gone. There are lots of real and concrete constraints (which seem to increase over time) on any physical system, making that finite data can reasonably approximate some of these systems very well. In the setting of actual infinity, things are different: finite is then just a trivial sample of the truly infinitely many possibilities.
In other words, the potential infinite in physics, is often quite approximable by finite means, because of the loads of restrictions present on physical systems.
> And as soon as you condone an attitude of dogmatic allegiance to a physical theory, there's nothing to stop dogmatic allegiance to silly green-cheese theories as well.
As mentioned above, please provide an example where the theory was wrong, but the experiment was discarded, because the
conclusion was "ZFC is consistent" or something similar.
Best,
Sam Sanders